Abstract

The Bayesian update can be viewed as a variational problem by characterizing the posterior as the minimizer of a functional. The variational viewpoint is far from new and is at the heart of popular methods for posterior approximation. However, some of its consequences seem largely unexplored. We focus on the following one: defining the posterior as the minimizer of a functional gives a natural path towards the posterior by moving in the direction of steepest descent of the functional. This idea is made precise through the theory of gradient flows, allowing to bring new tools to the study of Bayesian models and algorithms. Since the posterior may be characterized as the minimizer of different functionals, several variational formulations may be considered. We study three of them and their three associated gradient flows. We show that, in all cases, the rate of convergence of the flows to the posterior can be bounded by the geodesic convexity of the functional to be minimized. Each gradient flow naturally suggests a nonlinear diffusion with the posterior as invariant distribution. These diffusions may be discretized to build proposals for Markov chain Monte Carlo (MCMC) algorithms. By construction, the diffusions are guaranteed to satisfy a certain optimality condition, and rates of convergence are given by the convexity of the functionals. We use this observation to propose a criterion for the choice of metric in Riemannian MCMC methods.

Highlights

  • In this paper we revisit the old idea of viewing the posterior as the minimizer of an energy functional

  • The invariant distribution of each of these diffusions is the sought posterior, and a bound on the rate of convergence of the diffusions to the posterior is given by the geodesic convexity of the corresponding functional

  • The main contribution of this paper is to explore three variational formulations of the Bayesian update and their associated gradient flows

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Summary

Introduction

In this paper we revisit the old idea of viewing the posterior as the minimizer of an energy functional. The use of variational formulations of Bayes rule seems to have been largely focused on one of its methodological benefits: restricting the minimization to a subclass of measures is the backbone of variational Bayes methods for posterior approximation. Our aim is to bring attention to two other theoretical and methodological benefits, and to study in some detail one of these: namely, that each variational formulation suggests a natural path, defined by a gradient flow, towards the posterior. We use this observation to propose a criterion for the choice of metric in Riemannian MCMC methods. Given a prior p(u) on an unknown parameter u and a likelihood function L(y|u), the posterior p(u|y) ∝

30 The Bayesian Update
32 The Bayesian Update
Comparison of Functionals and Flows
Outline
Geodesic Spaces and Geodesic Convexity of Functionals
Gradient Flows in Metric Spaces
Variational Characterizations of the Posterior and Gradient Flows
Variational Formulation of the Bayesian Update
Geodesic Convexity and Functional Inequalities
PDEs and Diffusions
Application
Example
Probit and Logistic Models
Ginzburg-Landau Model
Conclusions and Future Work
Full Text
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